The axion arises as a pseudo Nambu-Goldstone boson from the spontaneous breaking of a hypothetical global Peccei-Quinn (PQ) symmetry introduced to provide a solution to the strong CP problem of quantum chromodynamics. Due to the weakness of the coupling with ordinary matters, the axion is regarded as a viable candidate of dark matter of the universe. However, the estimation of the axion dark matter abundance is not so straightforward if we follow the evolution of the axion field in the context of inflationary cosmology. In particular, if the PQ symmetry is restored after inflation, topological defects such as strings and domain walls are formed, and they produce significant amounts of cold axions. As a result, the prediction for the mass of axion dark matter depends strongly on the early history of the universe. In this talk, I will review recent developments of the theoretical estimation of the axion dark matter abundance in the post-inflationary PQ symmetry breaking scenario.